The least‐squares finite element method for low‐mach‐number compressible viscous flows

The least-squares finite element method (LSFEM), which is based on minimizing the l2-norm of the residual, has many attractive advantages over Galerkin finite element method (GFEM). It is now well established as a proper approach to deal with the convection dominated fluid dynamic equations. The least-squares finite element method has a number of attractive characteristics such as the lack of an inf-sup condition and the resulting symmetric positive system of algebraic equations unlike GFEM. However, the higher continuity requirements for second-order terms in the governing equations force the introduction of additional unknowns through the use of an equivalent first-order system of equations or the use of C1 continuous basis functions. These additional unknowns lead to increased memory and computing time requirements that have prevented the application of LSFEM to large-scale practical problems, such as three-dimensional compressible viscous flows. A simple finite element method is proposed that employs a least-squares method for first-order derivatives and a Galerkin method for second order derivatives, thereby avoiding the need for additional unknowns required by pure a LSFEM approach. When the unsteady form of the governing equations is used, a streamline upwinding term is introduced naturally by the leastsquares method. Resulting system matrix is always symmetric and positive definite and can be solved by iterative solvers like pre-conditioned conjugate gradient method. The method is stable for convection-dominated flows and allows for equalorder basis functions for both pressure and velocity. The stability and accuracy of the method are demonstrated with preliminary results of several benchmark problems solved using low-order C0 continuous elements.

Comparison of numerical methods for the solution of viscous incompressible and low Mach number compressible flow

The present paper reports the development of the least-squares finite-element method for simulating compressible buoyant flows at low Mach numbers. We propose a div-curl-grad formulation with unknowns including vorticity, velocity, heat fluxes, temperature and pressure variation. The formulation is proved to be elliptic such that permissible boundary conditions become self-evident for a well posed flow problem. In contrast to conventional approaches, the present method evades the predicament of the ‘singularity’ problem of low-speed flows and no special treatment or artificial boundary condition is needed. Moreover, the assembled coefficient matrix is symmetric and positive-definite; its inversion is implemented by an element-by-element jacobi conjugate gradient method. As a numerical example, we calculate two-dimensional compressible buoyant flows inside a square enclosure at various Rayleigh numbers. For Rayleigh number one million, four secondary vortices were found embedded in the primary vortex. Due to significant temperature variations, the fluid flows are highly compressible in the interior. Along the walls, however, the flows are incompressible. The Nusselt number-Rayleigh number correlation deduced from the numerical result compared favorably with previously reported data.

In this paper we devise a novel least-squares finite element method (LSFEM) for solving scalar convection-dominated convection-diffusion problems. First, we convert a second-order convection-diffusion problem into a first-order system formulation by introducing the gradient $\mathbf{p}:=-\kappa\nabla u$ as an additional variable. The LSFEM using continuous piecewise linear elements enriched with residual-free bubbles for both variables $u$ and $\mathbf{p}$ is applied to solve the first-order mixed problem. The residual-free bubble functions are assumed to strongly satisfy the associated homogeneous second-order convection-diffusion equations in the interior of each element, up to the contribution of the linear part, and vanish on the element boundary. To implement this two-level least-squares approach, a stabilized method of Galerkin/least-squares type is used to approximate the residual-free bubble functions. This enriched LSFEM not only inherits the advantages of the primitive LSFEM, such as the resulting linear system being symmetric and positive definite, but also exhibits the characteristics of the residual-free bubble method without involving stability parameters. Several numerical experiments are given to demonstrate the effectiveness of the proposed enriched LSFEM. The accuracy and computational cost of this enriched LSFEM are also compared with those of the primitive LSFEM. We find that for a small diffusivity $\kappa$, the enriched LSFEM is much better able to capture the nature of layer structure in the solution than the primitive LSFEM, even if the primitive LSFEM uses a very fine mesh or higher-order elements. In other words, the enriched LSFEM provides a significant improvement, with a lower computational cost, over the primitive LSFEM for solving convection-dominated problems.

This paper compares the performances of a least squares finite element method (LSFEM), a mixed Galerkin finite element method (MGFEM), and an implicit finite difference method (IFDM), when they are applied to an acoustic scattering problem modelled by the one‐dimensional Helmholtz equation. First, the boundary value problem is written as a first order system of differential equations, and variational formulations for the LSFEM and MGFEM are derived. Then, by using appropriate basis functions, the stiffness matrices and load vectors which define the corresponding linear algebraic systems are obtained. In all cases, the stiffness matrix is banded and for the LSFEM it is also Hermitian. Numerical tests show that all these methods have quadratic convergence to the exact solution. However, their efficiency assessed in terms of the number of nodes and computing time required to reach quadratic convergence varies. It is observed that LSFEM requires many more nodes and employs much more time than MGFEM and IFDM to reach the quadratic convergence for high frequencies. It is also found that MGFEM has less numerical dispersion and, as a consequence, performs better than IFDM and LSFEM for high frequencies. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A deficiency problem of the least squares finite element method for solving radiative transfer in strongly inhomogeneous media

Equivalence conditions for the finite volume and finite element methods in spherical coordinates

A high-order accurate unstructured spectral difference lattice Boltzmann method for computing inviscid and viscous compressible flows

In the present work, a mixed least‐squares finite element method (LSFEM) is used, which is applied on the partial differential equations arising from the Theory of Porous Media (TPM), see for example [1, 2, 3]. Since the LSFEM is not limited to the LBB condition, the method has some theoretical advantages over the well‐known (mixed) Galerkin method, see [4]. In particular, the LSFEM leads to positive definite and symmetric system matrices, even for differential equations with non‐self‐adjoint operators. In detail, we study an incompressible binary model with solid and liquid phases. The modeling of saturated porous structures serves as the basis for the presented approach. The resulting finite element is a four‐field formulation in terms of solid displacements, fluid pressure, mixture stresses, and a new variable associated with the pressure gradient. Vector‐valued Raviart‐Thomas and conventional Lagrangian functions are used to achieve a conforming discretization of the unknowns in the spaces H(div) and H1. The applicability of the LSFEM approach is demonstrated with numerical examples for fluid‐saturated porous structures, which consider incompressible, geometrically nonlinear elastic material behavior.

SUMMARYIn this paper, we develop least‐squares finite element methods (LSFEMs) for incompressible fluid flows with improved mass conservation. Specifically, we formulate a new locally conservative LSFEM for the velocity–vorticity–pressure Stokes system, which uses a piecewise divergence‐free basis for the velocity and standard C0 elements for the vorticity and the pressure. The new method, which we term dV‐VP improves upon our previous discontinuous stream‐function formulation in several ways. The use of a velocity basis, instead of a stream function, simplifies the imposition and implementation of the velocity boundary condition, and eliminates second‐order terms from the least‐squares functional. Moreover, the size of the resulting discrete problem is reduced because the piecewise solenoidal velocity element is approximately one‐half of the dimension of a stream‐function element of equal accuracy. In two dimensions, the discontinuous stream‐function LSFEM [1] motivates modification of our functional, which further improves the conservation of mass. We briefly discuss the extension of this modification to three dimensions. Computational studies demonstrate that the new formulation achieves optimal convergence rates and yields high conservation of mass. We also propose a simple diagonal preconditioner for the dV‐VP formulation, which significantly reduces the condition number of the LSFEM problem. Published 2012. This article is a US Government work and is in the public domain in the USA.

Enhanced anisotropic block-based adaptive mesh refinement for three-dimensional inviscid and viscous compressible flows

A new stabilized finite element formulation for solving radiative transfer equation is presented. It owns the salient feature of least-squares finite element method (LSFEM), i.e., free of the tuning parameter that appears in the streamline upwind/Petrov–Galerkin (SUPG) finite element method. The new finite element formulation is based on a second-order form of the radiative transfer equation. The second-order term will provide essential diffusion as the artificial diffusion introduced in traditional stabilized schemes to ensure stability. The performance of the new method was evaluated using challenging test cases featuring strong medium inhomogeneity and large gradient of radiative intensity field. It is demonstrated to be computationally efficient and capable of solving radiative heat transfer in strongly inhomogeneous media with even better accuracy than the LSFEM, and hence a promising alternative finite element formulation for solving complex radiative transfer problems.

A simple distribution function-based gas-kinetic scheme for simulation of viscous incompressible and compressible flows

Efficient stress–velocity least-squares finite element formulations for the incompressible Navier–Stokes equations

In this paper the simulation of incompressible flow in two dimensions by the least-squares finite element method (LSFEM) in the vorticity-velocity-pressure version is studied. In the LSFEM, the equations for continuity of mass and momentum and a vorticity equation are minimized on a discretization of the domain of interest. A problem is these equations are minimized in a global sense. Thus this method may not enforce that div$\underline{u} = 0$ at every point of the discretization. In this paper a modified LSFEM is developed which enforces near zero residual of mass conservation, i.e., div$\underline{u}$ is nearly zero at every point of the discretization. This is accomplished by adding an extra restriction in the divergence-free equation through the Lagrange multiplier strategy. In this numerical method the inf-sup or say LBB condition is not necessary, and the matrix resulting from applying the method on a discretization is symmetric; the uniqueness of the solution and the application of the conjugate gradient method are also valid. Numerical experience is given in simulating the flow of a cylinder with diameter 1 moving in a narrow channel of width 1.5. Results obtained by the LSFEM show that mass is created or destroyed at different points in the interior of discretization. The results obtained by the modified LSFEM show the mass is nearly conserved everywhere.